Magnetic Force-Free Theory: Nonlinear Case

In this paper, a theory of force-free magnetic field useful for explaining the formation of convex closed sets, bounded by a magnetic separatrix in the plasma, is developed. This question is not new and has been addressed by many authors. Force-free magnetic fields appear in many laboratory and astrophysical plasmas. These fields are defined by the solution of the problem ∇×B=ΛB with some field conditions B∂Ω on the boundary ∂Ω of the plasma region. In many physical situations, it has been noticed that Λ is not constant but may vary in the domain Ω giving rise to many different interesting physical situations. We set Λ=Λ(ψ) with ψ being the poloidal magnetic flux function. Then, an analytic method, based on a first-order expansion of ψ with respect to a small parameter α, is developed. The Grad–Shafranov equation for ψ is solved by expanding the solution in the eigenfunctions of the zero-order operator. An analytic expression for the solution is obtained deriving results on the transition through resonances, the amplification with respect to the gun inflow. Thus, the formation of Spheromaks or Protosphera structure of the plasma is determined in the case of nonconstant Λ.

Free-boundary plasma equilibria with toroidal plasma flows

Magnetohydrodynamic equilibrium schemes with toroidal plasma flows and the scrape-off layer are developed for the 'divertor-type' and 'limiter-type' free boundaries in the tokamak cylindrical coordinator. With a toroidal plasma flow, the flux functions are considerably different under the isentropic and isothermal assumptions. The effects of the toroidal flow on the magnetic axis shift are investigated. In a high beta plasma, the magnetic shift due to the toroidal flow are almost the same for both the isentropic and isothermal cases, and are about 0.04a0 (a0 is the minor radius) for M0=0.2 (the toroidal Alfvѐn Mach number on the magnetic axis). In addition, the X-point is slightly shifted upward by 0.0125 a0. But the magnetic axis and the X-point shift due to the toroidal flow may be neglected because M0 is usually less than 0.05 in a real tokamak. The effects of the toroidal flow on the plasma parameters are also investigated. The high toroidal flow shifts the plasma outward due to the centrifugal effect. Temperature profiles are noticeable different because the plasma temperature is a flux function in the isothermal case.

Network models for nonlocal traffic flow

We present a network formulation for a traffic flow model with nonlocal velocity in the flux function. The modeling framework includes suitable coupling conditions at intersections to either ensure maximum flux or distribution parameters. In particular, we focus on 1-to-1, 2-to-1 and 1-to-2 junctions. Based on an upwind type numerical scheme, we prove the maximum principle and the existence of weak solutions on networks. We also investigate the limiting behavior of the proposed models when the nonlocal influence tends to infinity. Numerical examples show the difference between the proposed coupling conditions and a comparison to the Lighthill-Whitham-Richards network model.

Small-scale Magnetic Flux Ropes and Their Properties Based on In Situ Measurements from the Parker Solar Probe

We report small-scale magnetic flux ropes via the in situ measurements from the Parker Solar Probe during the first six encounters, and present additional analyses to supplement our prior work in Chen et al. These flux ropes are detected by the Grad–Shafranov-based algorithm, with their durations and scale sizes ranging from 10 s to ≲1 hr and from a few hundred kilometers to 10−3 au, respectively. They include both static structures and those with significant field-aligned plasma flows. Most structures tend to possess large cross helicity, while the residual energy is distributed over wide ranges. We find that these dynamic flux ropes mostly propagate in the antisunward direction relative to the background solar wind, with no preferential signs of magnetic helicity. The magnetic flux function follows a power law and is proportional to scale size. We also present case studies showing reconstructed two-dimensional (2D) configurations, which confirm that both the static and dynamic flux ropes have a common configuration of spiral magnetic field lines (also streamlines). Moreover, the existence of such events hints at interchange reconnection as a possible mechanism for generating flux rope-like structures near the Sun. Lastly, we summarize the major findings, and discuss the possible correlation between these flux rope-like structures and turbulence due to the process of local Alfvénic alignment.

Green function for the Grad-Shafranov operator

The Grad-Shafranov equation, often written in cylindrical coordinates, is an elliptic partial differential equation in two dimensions. It describes magnetohydrodynamic equilibria in axisymmetric toroidal plasmas, such as tokamaks, and yields the poloidal magnetic flux function, which is related to the azimuthal component of the vector potential for the magnetic field produced by a circular (toroidal) current density. The Green function for the differential operator can be obtained from the vector potential for the magnetic field of a circular current loop, which is a typical problem in magnetostatics. The purpose of the paper is to collect results scattered in electrodynamics and plasma physics textbooks for the benefit of students in the field, as well as attracting the attention of a wider audience, in the context of electrodynamics and partial differential equations.

An equilibrated a posteriori error estimator for an Interior Penalty Discontinuous Galerkin approximation of the p-Laplace problem

We are concerned with an Interior Penalty Discontinuous Galerkin (IPDG) approximation of the p-Laplace equation and an equilibrated a posteriori error estimator. The IPDG method can be derived from a discretization of the associated minimization problem involving appropriately defined reconstruction operators. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W 1,p norm and relies on the construction of an equilibrated flux in terms of a numerical flux function associated with the mixed formulation of the IPDG approximation. The relationship with a residual-type a posteriori error estimator is established as well. Numerical results illustrate the performance of both estimators.

Capillary hysteresis and gravity segregation in two phase flow through porous media

We study the gravity driven flow of two fluid phases in a one dimensional homogeneous porous column when history dependence of the pressure difference between the phases (capillary pressure) is taken into account. In the hyperbolic limit, solutions of such systems satisfy the Buckley-Leverett equation with a non-monotone flux function. However, solutions for the hysteretic case do not converge to the classical solutions in the hyperbolic limit in a wide range of situations. In particular, with Riemann data as initial condition, stationary shocks become possible in addition to classical components such as shocks, rarefaction waves and constant states. We derive an admissibility criterion for the stationary shocks and outline all admissible shocks. Depending on the capillary pressure functions, flux function and the Riemann data, two cases are identified a priori for which the solution consists of a stationary shock. In the first case, the shock remains at the point where the initial condition is discontinuous. In the second case, the solution is frozen in time in at least one semi-infinite half. The predictions are verified using numerical results.

GKS and UGKS for High-Speed Flows

The gas-kinetic scheme (GKS) and the unified gas-kinetic scheme (UGKS) are numerical methods based on the gas-kinetic theory, which have been widely used in the numerical simulations of high-speed and non-equilibrium flows. Both methods employ a multiscale flux function constructed from the integral solutions of kinetic equations to describe the local evolution process of particles’ free transport and collision. The accumulating effect of particles’ collision during transport process within a time step is used in the construction of the schemes, and the intrinsic simulating flow physics in the schemes depends on the ratio of the particle collision time and the time step, i.e., the so-called cell’s Knudsen number. With the initial distribution function reconstructed from the Chapman–Enskog expansion, the GKS can recover the Navier–Stokes solutions in the continuum regime at a small Knudsen number, and gain multi-dimensional properties by taking into account both normal and tangential flow variations in the flux function. By employing a discrete velocity distribution function, the UGKS can capture highly non-equilibrium physics, and is capable of simulating continuum and rarefied flow in all Knudsen number regimes. For high-speed non-equilibrium flow simulation, the real gas effects should be considered, and the computational efficiency and robustness of the schemes are the great challenges. Therefore, many efforts have been made to improve the validity and reliability of the GKS and UGKS in both the physical modeling and numerical techniques. In this paper, we give a review of the development of the GKS and UGKS in the past decades, such as physical modeling of a diatomic gas with molecular rotation and vibration at high temperature, plasma physics, computational techniques including implicit and multigrid acceleration, memory reduction methods, and wave–particle adaptation.

High-Order Accurate Flux-Splitting Scheme for Conservation Laws with Discontinuous Flux Function in Space

A new modified Engquist–Osher-type flux-splitting scheme is proposed to approximate the scalar conservation laws with discontinuous flux function in space. The fact that the discontinuity of the fluxes in space results in the jump of the unknown function may be the reason why it is difficult to design a high-order scheme to solve this hyperbolic conservation law. In order to implement the WENO flux reconstruction, we apply the new modified Engquist–Osher-type flux to compensate for the discontinuity of fluxes in space. Together the third-order TVD Runge–Kutta time discretization, we can obtain the high-order accurate scheme, which keeps equilibrium state across the discontinuity in space, to solve the scalar conservation laws with discontinuous flux function. Some examples are given to demonstrate the good performance of the new high-order accurate scheme.